Since the OP is asking about generation ships as the preferred method for exploring the cosmos, it is possible to apply a combination of logic and imagination to work out the maximum speed of a generation ship.
The nearest planetary system is Proxima Centauri which is 4.243 light years distant. Assuming a generation ship will arrive at Proxima Centauri when the next generation is old enough to explore the system. If the average age of the primary generation is, say, thirty years. They will old enough and well trained enough to run the ship on its voyage. They will raise their children to also run the ship and be able to explore another planetary system. Therefore, it's not unreasonable to assume this second generation will have an average age of, say, thirty. The first generation will now be on average, sixty years plus.
This establishes our generation ship has travelled to Proxima Centauri in thirty years at a minimum. It's possible to add a few years, but at 30 years flight time this gives the fastest possible passage of a journey to Proxima Centauri.
This yields an average velocity of 14.143 percent of lightspeed (psol) or 42,430 km/s. Now this calculation neglects acceleration time. Therefore, the actual maximum velocity will be higher than the average. If this was pushed up to, say, fifteen (15) percent of lightspeed, then the time for acceleration can be subsumed into the average velocity.
There are numerous discussion of interstellar travel that assume this will only take place at velocities of one percent of lightspeed (3,000 km/s). This is based on factors concerning the amount of energy required to accelerate a vehicle to close to lightspeed and the mass rations involved. All of which are prohibitive.
At one psol, a generation ship take 424.3 years to reach Proxima Centauri. But allowing for an acceleration phase at one centimetre per second squared which it take ten years to attain one psol plus a further deceleration phase at its destination, this add an additional decade flight time. This means a total trip time of 434.3 years.
In summary, the maximum speed for a generation ship to travel to Proxima Centauri, our nearest planetary system, wi1l be of the order of fifteen psol or 0.15 c. While more realistic generation ships can be expected to have velocities around one psol and to take more generations to reach their destination.
The unit of psol (percentage of the speed of light) was devised by James Strong in his book, Flight to the Stars: An Inquiry into the Feasibility of Interstellar Flight (1963).